When putting in the sample does it need to be oozing out of the sides and the like to make proper contact with the upper and lower plates? Do I need to make sure the white border on the plates is clean and there is no DNA on it?

You don't need quite that much sample. You should ideally have enough to cover as much of the "inner field" as possible. But the main thing is to have enough sample to create a good dielectric for the capacitor that the plates form.

It's not going to make a lot of difference if a little sample gets on the border.